Special {base} | R Documentation

## Special Functions of Mathematics

### Description

Special mathematical functions related to the beta and gamma functions.

### Usage

    beta(a, b)
    lbeta(a, b)
    gamma(x)
    lgamma(x)
    digamma(x)
    trigamma(x)
    tetragamma(x)
    pentagamma(x)
    choose(n, k)
    lchoose(n, k)

### Arguments

`a, b, x` | numeric vectors.
`n, k` | integer vectors.

### Details

The functions `beta` and `lbeta` return the beta function and the natural logarithm of the beta function,

B(a,b) = (Gamma(a)Gamma(b))/(Gamma(a+b)).

The functions `gamma` and `lgamma` return the gamma function Γ(x) and the natural logarithm of the absolute value of the gamma function.

The functions `digamma`, `trigamma`, `tetragamma` and `pentagamma` return the first, second, third and fourth derivatives of the logarithm of the gamma function.

`digamma(x)` = psi(x) = d/dx {ln Gamma(x)} = Gamma'(x) / Gamma(x)

The functions `choose` and `lchoose` return binomial coefficients and their logarithms.

### References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole. (for `gamma` and `lgamma`.)

Abramowitz, M. and Stegun, I. A. (1972) Handbook of Mathematical Functions. New York: Dover. Chapter 6: Gamma and Related Functions.

### See Also

`Arithmetic` for simple, `sqrt` for miscellaneous mathematical functions and `Bessel` for the real Bessel functions.

### Examples

    choose(5, 2)
    for (n in 0:10) print(choose(n, k = 0:n))

    ## gamma has discontinuities are 0, -1, -2, ...
    ## Don't run:
    ## use plots of points to show this.
    curve(gamma(x),-3,4, n=1001, ylim=c(-10,100),
          col="red", lwd=2, main="gamma(x)")
    abline(h=0,v=0, lty=3, col="midnightblue")
    ## End Don't run

    x <- seq(.1, 4, length = 201); dx <- diff(x)[1]
    par(mfrow = c(2, 3))
    for (ch in c("", "l","di","tri","tetra","penta")) {
      is.deriv <- nchar(ch) >= 2
      if (is.deriv) dy <- diff(y) / dx
      nm <- paste(ch, "gamma", sep = "")
      y <- get(nm)(x)
      plot(x, y, type = "l", main = nm, col = "red")
      abline(h = 0, col = "lightgray")
      if (is.deriv) lines(x[-1], dy, col = "blue", lty = 2)
    }

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